Knot theory

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Knot theory is a very fast growing field of mathematics. Knots are not natural phenomena, and there exists only a finite number of distinct knots in three-dimensional space. Knots define spaces, because we can think of a knot as a way in which different dimensions can be connected. Mathematicians are working on notation systems for knots, which leads to a form of arithmetic for knots. This has fascinating consequences for other disciplines, and for our understanding of reality in general (see the heading “Virtual knots”.)

What is a knot?

Complex knots can oftentimes be simplified with a few moves, which the German mathematician Reidemeister organized into three categories. (Reidemeister moves.) Once the knot is simplified and no further crossing can be removed, the knot is classified by the number of crossings that remain. This is called a “prime knot.” For example, the trefoil knot is classified by its fewest number of crossings – three (see the diagram below).

It is possible to have more than one prime knot with the same number of crossings. In this case, we usually use subscripts to denote different knots with the same number of crossings, such as the 51 and 52 knots in the diagram below:

There is no known formula for giving the number of distinct prime knots as a function of the number of crossings. The numbers of distinct prime knots having n= 1, 2, 3,… crossings are 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988.

Virtual Knots?

The following quote gives first a short introduction to knots, and then it discusses the discovery of virtual knots. From: A revolution in knot theory (physorg.com). Provided by American Mathematical Society.

“As sailors have long known, many different kinds of knots are possible; in fact, the variety is infinite. A *mathematical* knot can be imagined as a knotted circle: Think of a pretzel, which is a knotted circle of dough, or a rubber band, which is the “un-knot” because it is not knotted. Mathematicians study the patterns, symmetries, and asymmetries in knots and develop methods for distinguishing when two knots are truly different.

Mathematically, one thinks of the string out of which a knot is formed as being a one-dimensional object, and the knot itself lives in three-dimensional space. Drawings of knots, like the ones done by Tait, are projections of the knot onto a two-dimensional plane. In such drawings, it is customary to draw over-and-under crossings of the string as broken and unbroken lines. If three or more strands of the knot are on top of each other at single point, we can move the strands slightly without changing the knot so that every point on the plane sits below at most two strands of the knot. A planar knot diagram is a picture of a knot, drawn in a two-dimensional plane, in which every point of the diagram represents at most two points in the knot. Planar knot diagrams have long been used in mathematics as a way to represent and study knots.

As Nelson reports in his article, mathematicians have devised various ways to represent the information contained in knot diagrams. One example is the Gauss code, which is a sequence of letters and numbers wherein each crossing in the knot is assigned a number and the letter O or U, depending on whether the crossing goes over or under. The Gauss code for a simple knot might look like this: O1U2O3U1O2U3.

In the mid-1990s, mathematicians discovered something strange. There are Gauss codes for which it is impossible to draw planar knot diagrams but which nevertheless behave like knots in certain ways. In particular, those codes, which Nelson calls *nonplanar Gauss codes*, work perfectly well in certain formulas that are used to investigate properties of knots. Nelson writes: “A planar Gauss code always describes a [knot] in three-space; what kind of thing could a nonplanar Gauss code be describing?” As it turns out, there are “virtual knots” that have legitimate Gauss codes but do not correspond to knots in three-dimensional space. These virtual knots can be investigated by applying combinatorial techniques to knot diagrams.

Just as new horizons opened when people dared to consider what would happen if -1 had a square root—and thereby discovered complex numbers, which have since been thoroughly explored by mathematicians and have become ubiquitous in physics and engineering—mathematicians are finding that the equations they used to investigate regular knots give rise to a whole universe of “generalized knots” that have their own peculiar qualities. Although they seem esoteric at first, these generalized knots turn out to have interpretations as familiar objects in mathematics. “Moreover,” Nelson writes, “classical knot theory emerges as a special case of the new generalized knot theory.”

Knot Tables

When we consider all the available knots up to 9 crossings, we get the following table with this system of notation: